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Euclid: Elementa
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PROPOSITION 27. 
 
 
Similar solid numbers have to one another the ratio which a cube number has to a cube number. 
 
 
Let A, B be similar solid numbers;  I say that A has to B the ratio which a cube number has to a cube number.  For, since A, B are similar solid numbers, therefore two mean proportional numbers fall between A, B. [VIII. 19]  Let C, D so fall, and let E, F, G, H, the least numbers of those which have the same ratio with A, C, D, B, and equal with them in multitude, be taken; [VII. 33 or VIII. 2]  therefore the extremes of them E, H are cube. [VIII. 2, Por.]  And, as E is to H, so is A to B;  therefore A also has to B the ratio which a cube number has to a cube number.  Q. E. D. 
               
               
BOOK IX. 
 
 
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